Optimal cooperative search in fractional cascaded data structures

Fractional cascading is a technique designed to allow efficient sequential search in a graph with catalogs of total sizen. The search consists of locating a key in the catalogs along a path. In this paper we show how to preprocess a variety of fractional cascaded data structures whose underlying graph is a tree so that searching can be done efficiently in parallel. The preprocessing takesO(logn) time withn/logn processors on an EREW PRAM. For a balanced binary tree, cooperative search along root-to-leaf paths can be done inO((logn)/logp) time usingp processors on a CREW PRAM. Both of these time/processor constraints are optimal. The searching in the fractional cascaded data structure can be either explicit, in which the search path is specified before the search starts, or implicit, in which the branching is determined at each node. We apply this technique to a variety of geometric problems, including point location, range search, and segment intersection search.

[1]  Michael T. Goodrich,et al.  Triangulating a Polygon in Parallel , 1989, J. Algorithms.

[2]  Leonidas J. Guibas,et al.  Fractional cascading: II. Applications , 1986, Algorithmica.

[3]  David G. Kirkpatrick,et al.  Optimal Search in Planar Subdivisions , 1983, SIAM J. Comput..

[4]  Chee-Keng Yap.  Parallel triangulation of a polygon in two calls to the trapezoidal map , 2005, Algorithmica.

[5]  Leonidas J. Guibas,et al.  Fractional cascading: I. A data structuring technique , 1986, Algorithmica.

[6]  Jeffrey Scott Vitter,et al.  Parallel Transitive Closure and Point Location in Planar Structures , 1991, SIAM J. Comput..

[7]  Marc Snir,et al.  On Parallel Searching , 2011, SIAM J. Comput..

[8]  Stefan Näher.  Dynamic Fractional Cascading oder die Verwaltung vieler linearer Listen , 1987 .

[9]  Michael T. Goodrich,et al.  Planar Separators and Parallel Polygon Triangulation , 1995, J. Comput. Syst. Sci..

[10]  Kenneth L. Clarkson,et al.  Randomized parallel algorithms for trapezoidal diagrams , 1991, SCG '91.

[11]  Bernard Chazelle.  How to Search in History , 1983, FCT.

[12]  Takao Asano,et al.  A new point-location algorithm and its practical efficiency: comparison with existing algorithms , 1984, TOGS.

[13]  Kenneth L. Clarkson,et al.  Erratum: Randomized parallel algorithms for trapezoidal diagrams , 1992, International journal of computational geometry and applications.

[14]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[15]  Kenneth L. Clarkson,et al.  Randomized parallel algorithms for trapezoidal diagrams , 1992, Int. J. Comput. Geom. Appl..

[16]  D. T. Lee,et al.  Location of a point in a planar subdivision and its applications , 1976, STOC '76.

[17]  Richard Cole,et al.  Cascading Divide-and-Conquer: A Technique for Designing Parallel Algorithms , 1987, FOCS.

[18]  Leonidas J. Guibas,et al.  Optimal Point Location in a Monotone Subdivision , 1986, SIAM J. Comput..

[19]  Michael T. Goodrich,et al.  Planar separators and parallel polygon triangulation (preliminary version) , 1992, STOC '92.